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Finance rate guide

APR vs APY

APR describes borrowing cost under a credit disclosure. APY describes annual deposit yield with compounding. Learn which rate belongs in a calculation, then compare the actual dollar amounts.

Worked examples · Three original charts · US disclosure context

Try the worked APY example

APR, APY and nominal interest rate are different terms

Annual percentage rate, or APR, is used in borrowing disclosures. Annual percentage yield, or APY, is used to express deposit-interest growth over a year under stated assumptions. A nominal annual interest rate is the rate before its specified periodic compounding is applied. Identifying these three concepts prevents a common mistake: treating a loan APR that includes finance charges as though it were a savings account's nominal interest rate.

The disclosure discussion here follows US consumer banking sources. Other jurisdictions may define or label rates differently. All example amounts are in US dollars, and all example rates are hypothetical rather than current product offers.

Identify the measure before comparing percentages
TermWhat it measuresWhat to check
APRAnnual borrowing cost under the applicable credit-product rules.Included finance charges, payment timing, term and the lender's disclosure. Mortgage APR can include costs beyond interest.
Nominal annual interest rateA quoted annual interest rate paired with a periodic convention.The compounding schedule and whether the input is actually a nominal rate.
APYAnnualized deposit-interest yield that includes compounding.Rate changes, balance tiers, fees, term conditions and whether interest stays deposited.

On mobile, swipe wide tables or focus them and use the arrow keys. Browser Print can save a reference copy as PDF; expand any practice answers you want included.

The CFPB mortgage explanation distinguishes the interest rate from APR. Its deposit APY rules address the annualized interest yield. These are related financial ideas, but the labels do not make them interchangeable inputs.

Comparison of borrowing APR and deposit APY, distinguishing credit costs from compounded savings yield.
APR is a borrowing-cost disclosure; APY is an annual deposit-interest yield. A nominal interest rate is a separate input for compounding calculations.

The nominal-rate-to-APY formula

Let rr be a nominal annual interest rate as a decimal, nn the number of equal compounding periods in one year, PP the starting deposit, and qq the effective annual yield as a decimal. For example, 6%=0.066\%=0.06, not 66, is the value to use for rr.

q=(1+rn)n−1q=\left(1+\frac{r}{n}\right)^n-1

One-year ending balance

A=P(1+q)=P(1+rn)nA=P(1+q)=P\left(1+\frac{r}{n}\right)^n

One-year interest

I=A−P=PqI=A-P=Pq

The model keeps the rate fixed, retains the interest, and has no additional deposits, withdrawals or fees. It uses equal compounding periods and no intermediate cents rounding. Actual account calculations can use calendar dates, balance rules and posting conventions that differ from this idealized model.

For periodic compounding, nn must be positive and 1+r/n>01+r/n>0. The examples use nonnegative rates. With a positive nominal rate and more than one compounding period, q>rq>r. With annual compounding, n=1n=1 and q=rq=r; at a zero rate, both are zero.

Worked example: monthly compounding

Start with P=10,000P=10{,}000 dollars, a fixed 6%6\% nominal annual rate and n=12n=12 monthly periods. Interest remains in the account for one year. Work from the periodic rate to the annual result.

Interest builds on retained interest diagramMonth-end interest on ten thousand dollars at six percent nominal interest compounded monthly. It reaches 616 dollars and 78 cents after twelve months. A dashed simple-interest reference reaches 600 dollars. Both axes begin at zero.Interest in US dollars
00
200200
400400
600600
00
33
66
99
1212
Completed months

Interest builds on retained interest

I12≈$616.78I_{12}\approx\$616.78Isimple=$600I_{\mathrm{simple}}=\$600

Dots are month-end values; solid segments connect those observations. The dashed line is a simple-interest accrual reference, not a second product offer or a daily crediting schedule. No intermediate cents rounding is applied.

1. Find the rate for one month

i=rn=0.0612=0.005=0.5%i=\frac{r}{n}=\frac{0.06}{12}=0.005=0.5\%

The first month's interest is 10,000(0.005)=5010{,}000(0.005)=50 dollars. The balance becomes 10,05010{,}050 dollars. The second month's interest is 10,050(0.005)=50.2510{,}050(0.005)=50.25 dollars, giving a balance of 10,100.2510{,}100.25 dollars. Interest on the retained first month's interest explains the extra amount.

2. Compound through the full year

q=(1.005)12−1≈0.0616778119q=(1.005)^{12}-1\approx0.0616778119APY⁡≈6.167781%\operatorname{APY}\approx6.167781\%

3. Convert the yield into dollars

A=10,000(1.005)12≈$10,616.78A=10{,}000(1.005)^{12}\approx\$10{,}616.78I=A−10,000≈$616.78I=A-10{,}000\approx\$616.78Reproduce the monthly example

Simple interest at the same nominal rate would be 10,000(0.06)=60010{,}000(0.06)=600 dollars for the year. The modeled compounding benefit is approximately 16.7816.78 dollars. APY is approximately 6.17%6.17\% if displayed to two decimal places; keep the unrounded growth factor when calculating the dollar result.

For deposit timing and longer projections, the Compound Interest Examples guide extends the same recurrence. More deposits change the cash-flow problem; they do not change the definition of the account's quoted APY.

Compare compounding frequencies on the same basis

Every row below uses the same 6%6\% nominal rate, 10,00010{,}000 opening deposit and one-year horizon. Only the compounding convention changes. These are mathematical comparisons, not rates advertised by different banks.

Same nominal rate, different effective yields
FrequencyPeriods per yearAPY, roundedOne-year interest
Annual116.000000%6.000000\%$600.00\$600.00
Semiannual226.090000%6.090000\%$609.00\$609.00
Quarterly446.136355%6.136355\%$613.64\$613.64
Monthly12126.167781%6.167781\%$616.78\$616.78
Daily, 365 periods3653656.183131%6.183131\%$618.31\$618.31
Continuous limitn→∞n\to\infty6.183655%6.183655\%$618.37\$618.37

The continuous-compounding limit is q=er−1q=e^r-1. It is a mathematical benchmark, not a claim that an account offers continuous posting. Daily compounding here means 365365 equal periods. A different day-count convention or a leap-year calculation needs its stated basis.

Compounding and crediting are not identical labels. Interest may be calculated using daily balances but credited on another schedule. The account disclosure requirements address both frequencies. A statement saying interest is paid monthly does not, by itself, establish a monthly compounding model.

Convert APY back to a nominal annual rate

Suppose a fixed account yield is 5%5\% APY and you need its equivalent monthly-compounded nominal rate. Start with q=0.05q=0.05 and n=12n=12. Undo the annual power before multiplying by the period count.

  1. Take the monthly growth factor: (1+q)1/12(1+q)^{1/12}.
  2. i=(1.05)1/12−1≈0.0040741238i=(1.05)^{1/12}-1\approx0.0040741238This is approximately 0.407412%0.407412\% per month.
  3. r=12[(1.05)1/12−1]≈0.0488894854r=12\left[(1.05)^{1/12}-1\right]\approx0.0488894854The equivalent nominal annual rate is approximately 4.888949%4.888949\%.
Check APY-to-nominal conversion

Avoid compounding APY a second time

Entering 5%5\% APY as a 5%5\% nominal rate and compounding monthly produces approximately 5.116190%5.116190\% effective yield. That is a different assumption. With the original 5%5\% APY held constant for one year, a 10,00010{,}000 unchanged deposit earns 500500 dollars. Changing the conversion frequency while keeping APY fixed does not create additional annual earnings.

Annualize a short-term return without changing the payout

An illustrative deposit of 10,00010{,}000 dollars earns 250250 dollars over exactly 180180 days. No other cash flows occur, and interest stays deposited. The term return is 250/10,000=0.025=2.5%250/10{,}000=0.025=2.5\%. To express it as a 365365-day annual-equivalent yield, use the actual term length dd in days.

q=(1+IP)365/d−1q=\left(1+\frac{I}{P}\right)^{365/d}-1q=(1.025)365/180−1≈0.0513458771q=(1.025)^{365/180}-1\approx0.0513458771

The annualized yield is approximately 5.134588%5.134588\%. The maturity balance is still 10,25010{,}250 dollars, with actual term interest of 250250 dollars. The larger annual-equivalent figure does not become an additional payment at maturity and does not guarantee reinvestment at the same rate.

Reproduce the actual-day annualization

The US deposit APY formula uses annualization and stated assumptions about the deposit term. It also contains special rules for certain products and permits a leap-year basis in the circumstances it describes. The worked calculation here declares its day basis explicitly.

Project a term balance from an already quoted APY

Under a constant equivalent-yield model, a different deposit at 5%5\% APY for 180180 days has the following projection. Use the annual yield directly instead of dividing it by a period count and compounding again.

A=10,000(1.05)180/365≈$10,243.53A=10{,}000(1.05)^{180/365}\approx\$10{,}243.53

Modeled interest is approximately 243.53243.53 dollars. This is a 180180-day projection, not a claim about every calendar six-month CD. Bank crediting rules, mandatory interest payouts and withdrawal conditions can change the actual cash received.

Check the CD projection with APY as the input

A higher gross APY can still leave less after fees

A fee can reverse a close comparison diagramThree zero-based bars: account A gross interest of 616 dollars and 78 cents, A after a separate 24-dollar year-end fee at 592 dollars and 78 cents, and fee-free account B at 615 dollars. Rates are illustrative and taxes are excluded.One-year gain in US dollars
00
200200
400400
600600
616.78616.78
A: gross
592.78592.78
A: after fee
615.00615.00
B: no fee

A fee can reverse a close comparison

The vertical axis starts at zero. A has the higher gross yield, but B retains more money after the stated fee. The fee is paid separately at year end, so it does not change the balance earning interest during the year.

Compare two hypothetical unchanged 10,00010{,}000 deposits for one year. Account A pays the 6%6\% nominal rate compounded monthly from our example. Account B pays a fixed 6.15%6.15\% APY with no fee. Before fees, A earns approximately 616.78616.78 dollars and B earns 615615 dollars.

1. Put both rates on an annual-yield basis

qA≈6.167781%,qB=6.15%q_A\approx6.167781\%,\qquad q_B=6.15\%

A's gross advantage is only about 1.781.78 dollars. Comparing the headline 6%6\% nominal rate directly with B's 6.15%6.15\% APY would miss that result.

2. Deduct the stated fee at its stated time

Now suppose A charges 2424 dollars separately at year end. This fee does not reduce the deposit balance during compounding. Taxes are excluded from both examples.

GA=10,000qA−24≈$592.78G_A=10{,}000q_A-24\approx\$592.78GB=10,000(0.0615)=$615.00G_B=10{,}000(0.0615)=\$615.00

3. Compare the retained dollar amounts

B now leaves approximately 22.2222.22 dollars more. A's net gain divided by its initial deposit is approximately 5.927781%5.927781\%. That is this scenario's after-fee return, not a replacement label for the account's disclosed APY.

Check B's gross APY result. The calculator link does not deduct A's fee; the separate arithmetic above shows that assumption explicitly. A monthly fee taken from the account would also reduce subsequent interest and requires a balance-by-balance model. Account fees and their conditions belong in the deposit disclosure.

Why a borrowing APR is not just a savings conversion

A credit APR depends on the relevant disclosure rules and the cash flows. For a mortgage, specified charges can make APR different from the contract interest rate. For closed-end credit, Regulation Z connects credit received and repayments through their amounts and timing. Open-end credit has its own periodic-rate rules.

Compare cash received with cash repaid diagramBorrower cash flows for an illustrative single-payment loan: receive 950 dollars at origination after a 50-dollar fee is withheld from a 1000-dollar face amount; repay 1100 dollars one year later. Positive is money received and negative is money repaid. Bar lengths use a common dollar scale.Borrower cash flow in US dollars
−1000-1000
00
10001000
+950+950
−1100-1100
At originationOne year later

Compare cash received with cash repaid

The common vertical scale shows money received above zero and money repaid below zero. This is a single-payment cash-flow illustration, not an amortizing-loan schedule or a certified regulatory APR disclosure.

A one-payment cash-flow illustration

Imagine a loan with a 1,0001{,}000 face amount, a 5050 upfront fee withheld from the advance, and a single repayment exactly one year later. The contract interest for that year is 10%10\% of the face amount. There are no intermediate payments or other charges in this illustration.

  1. Cash actually received is C=1,000−50=950C=1{,}000-50=950 dollars.
  2. Cash repaid is R=1,000(1+0.10)=1,100R=1{,}000(1+0.10)=1{,}100 dollars.
  3. a=R−CC=150950=319a=\frac{R-C}{C}=\frac{150}{950}=\frac3{19}a≈15.789474%a\approx15.789474\%

The annual cash-flow cost exceeds the contract interest rate because less cash was received at the start. This example isolates that effect. It is not a certified APR disclosure: real APR calculations must identify the applicable finance charges, payment dates and required method.

Credit cards need their own periodic convention

For a card using an 18%18\% annualized periodic rate and a 365365-day divisor, the daily rate is id=0.18/365≈0.0004931507i_d=0.18/365\approx0.0004931507. On a 1,0001{,}000 balance for one such day, that is about 0.490.49 dollars of interest. This single-day example does not determine a whole statement's charges.

The CFPB daily-rate explanation notes that issuer day conventions can differ. Read the card agreement for its balance method, compounding, fees, grace period and payment treatment. A savings APY formula alone cannot answer a card payoff question.

For monthly amortizing loans, use the Loan Payment Calculator with the contract interest rate and the actual fee treatment. Its payment projection is a different model from the single-payment example. The amortization guide explains why timing and declining principal matter.

Check these assumptions before using a result

  1. Identify the rate. Match the input label to a nominal rate, APY or a legally defined borrowing APR. Do not substitute one simply because each is expressed annually.
  2. Match the term and balance. Include the opening deposit, dates, subsequent cash flows and any balance tier. A quoted yield is not an exact cash payout for every holding period.
  3. Read the rate conditions. Fixed, variable and promotional products can have different assumptions. An introductory rate does not automatically apply to the entire year.
  4. Account for retained interest and charges. Withdrawn interest no longer compounds in the same account. Fees, penalties and taxes change the retained amount and need their own timing.
  5. Round last and retain the source. Keep the original rate precision for calculations and compare the result with the institution's disclosure and statement conventions.

Practice with answers

Try each question before opening its answer. These disclosures work without JavaScript.

What does a 4%4\% nominal rate produce with annual compounding?

There is one period, so q=(1+0.04)1−1=0.04=4%q=(1+0.04)^1-1=0.04=4\%. A 10,00010{,}000 deposit earns 400400 dollars under the stated constant-rate, no-transaction model. Annual compounding does not add within-year interest on interest; this equality refers to the nominal rate and yield, not to an unrelated fee-inclusive loan APR.

Does a zero nominal rate earn more when compounded daily?

No. Every growth factor remains one: (1+0/365)365−1=0(1+0/365)^{365}-1=0. More compounding periods cannot create interest from a zero rate. Fees could still reduce the amount retained, which would be a separate cash-flow effect rather than positive interest from compounding.

How much does an unchanged 10,00010{,}000 deposit earn at 5%5\% APY over one year?

With that APY maintained and no other transactions or fees, I=10,000(0.05)=500I=10{,}000(0.05)=500 dollars. The ending balance is 10,50010{,}500 dollars. APY already incorporates the annual growth effect. Treating it as a monthly-compounded nominal rate would answer a different question.

FAQ

Frequently Asked Questions

What is the difference between APR and APY?
APR is an annual borrowing-cost disclosure whose calculation depends on the credit product and applicable rules. For mortgages it can reflect specified charges beyond the interest rate. APY expresses an annual deposit-interest yield with compounding under stated assumptions. The nominal annual interest rate used in a compounding formula is a separate concept; treating every disclosed loan APR as that nominal rate can give a misleading comparison.
Is APY always higher than APR?
That comparison is meaningful only after identifying what both numbers measure. For the same positive nominal interest rate, with interest retained and more than one compounding period per year, the effective annual yield exceeds the nominal rate. Annual compounding or a zero rate makes those two model rates equal. A borrowing APR that includes finance charges is not that same nominal savings rate, so there is no universal ordering.
Can I convert any loan APR into APY?
No. The periodic-compounding formula needs a nominal annual interest rate and its matching periodic convention. A disclosed loan APR can reflect payment timing and specified finance charges, so inserting it into that formula does not reconstruct the loan's cash flows. Use the lender's APR disclosure to compare like borrowing products and a payment model with the actual contract rate, fees and schedule for cash amounts.
Does APY include account fees or taxes?
Do not treat deposit APY as your personal after-fee, after-tax return. Account fees are disclosed separately, and their timing can change the balance that earns interest. The article's comparison deducts a separate year-end fee without changing the year's deposit balance. A monthly deduction would need a transaction-by-transaction model instead. Taxes, withdrawals and penalties also change the amount you ultimately retain, so keep those effects separate from quoted APY.
Can I divide APY by twelve to find the monthly rate?
Not for an equivalent monthly compound rate. If qq is the annual yield as a decimal, the monthly rate is (1+q)1/12−1(1+q)^{1/12}-1. At 5%5\% APY this is approximately 0.407412%0.407412\% per month. Dividing a nominal annual rate by twelve is appropriate for its stated monthly convention, but doing that to APY and then compounding again counts growth incorrectly. Keep the two input types distinct.
Does an advertised APY guarantee that yield for the next year?
Only the product's actual terms can establish a fixed rate for a stated term. A variable account can change its rate, and a promotional offer can have different conditions after an introductory period. A quoted annual yield is not a promise that every future month will earn an unchanged rate. Check the account disclosure, applicable balance tier and rate-change terms before applying a constant-rate projection.
Does a short-term CD earn its full advertised APY before maturity?
APY is annualized, so it is not normally the interest percentage received over a shorter term. The worked example earns two and a half percent over its stated term and then annualizes that return for comparison. Its maturity interest remains the actual term interest, not the larger annual-equivalent figure. Exact maturity dates, day-count conventions, crediting requirements and any early-withdrawal penalty belong in a CD cash-flow estimate.
Does daily compounding always beat monthly compounding?
Holding a positive nominal annual rate and all other assumptions fixed, daily compounding gives a slightly higher effective annual yield than monthly compounding. Holding APY fixed instead, more frequent compounding corresponds to a lower nominal rate that produces the same annual growth. Real products can also differ in fees, balance requirements and rate tiers. Compare the stated yield and conditions together rather than treating frequency as an independent bonus.

Sources and calculation verification

Definitions are grounded in the primary US references below. The original examples use independently verified exact arithmetic or high-precision numerical evaluation. Chart coordinates, rate-conversion round trips and linked calculator settings are checked against those examples. Dollar displays round at the end; a real institution may round or credit interest differently.

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